Serre relations and Serre's theorem

Intuition [Beginner]

Serre's theorem says you can build an entire semisimple Lie algebra from a tiny set of generators and relations. The Cartan matrix $A$ gives you the recipe.

You need $3\ell$ generators: $\ell$ pairs of raising/lowering operators $e_i,f_i$ and $\ell$ Cartan generators $h_i$ (one for each simple root). Then you impose four types of relations:

1. The Cartan generators commute with each other: $[h_i,h_j]=0$.
2. The $e$'s and $f$'s are eigenvectors of the $h$'s: $[h_i,e_j]=A_{ij}{e}_j$ and $[h_i,f_j]=-A_{ij}{f}_j$.
3. Each $e_i$ and $f_i$ pair to give an $h$: $[e_i,f_j]={\delta }_{ij}{h}_i$.
4. The Serre relations: kill repeated brackets when $i\ne j$.

The magic of the Serre relations is that they prevent the Lie algebra from being too big. Without them, the generators would produce an infinite-dimensional algebra. The Serre relations trim it down to exactly the right finite-dimensional semisimple Lie algebra.

Visual [Beginner]

A diagram showing the three types of generators for $\mathfrak{sl}(3)$ (type $A_2$): $e_1,f_1,h_1$ and $e_2,f_2,h_2$. Arrows show the bracket relations between them. The Serre relation $(\text{ad }{e}_1{)}^2{e}_2=0$ is shown as a crossed-out arrow: "you cannot raise twice in direction 1 then once in direction 2."

The Serre relations: the constraints that keep the Lie algebra finite and exactly the right size.

Worked example [Beginner]

For $\mathfrak{sl}(3)$ (type $A_2$, Cartan matrix $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$):

The Serre relations are:

• $(\text{ad }{e}_1{)}^2{e}_2=[e_1,[e_1,e_2]]=0$
• $(\text{ad }{e}_2{)}^2{e}_1=[e_2,[e_2,e_1]]=0$
• Same for the $f$'s: $(\text{ad }{f}_1{)}^2{f}_2=0$ and $(\text{ad }{f}_2{)}^2{f}_1=0$.

The exponent $1-A_{ij}=1-(-1)=2$ for each pair. So "bracket twice" is forbidden.

Starting from $e_1$ and $e_2$, the only new element is $[e_1,e_2]$ (bracket once is allowed). Br $[e_1,[e_1,e_2]]$ is killed by the Serre relation. So the positive root space has dimension 3 (generated by $e_1,e_2,[e_1,e_2]$), matching the three positive roots of $A_2$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Chevalley-Serre presentation). Let $A=(A_{ij})$ be a Cartan matrix of rank $\ell$. Define the Lie algebra $\mathfrak{g}(A)$ with generators $\{e_i,f_i,h_i:1\le i\le \ell \}$ and relations:

(S1) $[h_i,h_j]=0$ for all $i,j$.

(S2) $[h_i,e_j]=A_{ij}{e}_j$ and $[h_i,f_j]=-A_{ij}{f}_j$ for all $i,j$.

(S3) $[e_i,f_j]={\delta }_{ij}{h}_i$ for all $i,j$.

(S4) $(\text{ad }{e}_i{)}^{1-A_{ij}}{e}_j=0$ for $i\ne j$.

(S5) $(\text{ad }{f}_i{)}^{1-A_{ij}}{f}_j=0$ for $i\ne j$.

Relations (S4-S5) are the Serre relations. They express the fact that the $\mathfrak{sl}(2)$-string through $e_j$ in the direction of $e_i$ has length at most $1-A_{ij}$.

Key theorem with proof [Intermediate+]

Theorem (Serre, 1966). Let $A$ be a Cartan matrix of finite type (positive definite). Then $\mathfrak{g}(A)$ is a finite-dimensional semisimple Lie algebra with Cartan matrix $A$. Conversely, every finite-dimensional semisimple Lie algebra arises this way.

Proof sketch. The proof proceeds in several stages.

Step 1: Well-definedness. Show that $\mathfrak{g}(A)$ is nontrivial by constructing a representation. The adjoint representation of a semisimple Lie algebra with Cartan matrix $A$ satisfies all five relations, so $\mathfrak{g}(A)$ maps onto it.

Step 2: The $\mathfrak{sl}(2)$-string argument. For each $i$, the triple $(e_i,h_i,f_i)$ generates a copy of $\mathfrak{sl}(2)$. The Serre relation $(\text{ad }{e}_i{)}^{1-A_{ij}}{e}_j=0$ says that the $\mathfrak{sl}(2{)}_i$-string through $e_j$ has length at most $-A_{ij}$. By the representation theory of $\mathfrak{sl}(2)$, this string must terminate, giving:

$$(\text{ad }{e}_i{)}^{1-A_{ij}}{e}_j=0,(\text{ad }{f}_i{)}^{1-A_{ij}}{f}_j=0.$$

These are exactly the Serre relations (S4-S5).

Step 3: Root system. From (S1-S3), the $h_i$ span a Cartan subalgebra, and the $e_i$ and $f_i$ are root vectors. The root system generated has Cartan matrix $A$.

Step 4: Finiteness. The positive-definiteness of $A$ forces the root system to be finite (the Weyl group is finite). Hence $\mathfrak{g}(A)$ is finite-dimensional.

Step 5: Semisimplicity. The Killing form restricted to the Cartan subalgebra is given by $B(h_i,h_j)=d_i{A}_{ij}$ where $d_i>0$. Since $A$ is positive definite, $B$ is nondegenerate, and Cartan's criterion gives semisimplicity. $\square$

Bridge. Serre's theorem converts the classification of semisimple Lie algebras into a combinatorial problem; the foundational reason is that the Cartan matrix provides a complete presentation by generators and relations. This pattern appears again in the theory of Kac-Moody algebras where the same presentation with an indefinite Cartan matrix produces infinite-dimensional Lie algebras. The bridge is that the Serre presentation identifies the Lie algebra with its Cartan matrix, converting geometry into algebra and back.

Exercises [Intermediate+]

Advanced results [Master]

Kac-Moody algebras. When the Cartan matrix $A$ is only assumed to be a generalised Cartan matrix ($A_{ii}=2$, $A_{ij}\le 0$ for $i\ne j$, $A_{ij}=0\iff A_{ji}=0$), the Serre presentation still defines a Lie algebra $\mathfrak{g}(A)$. If $A$ is positive semidefinite with corank $1$, the result is an affine Kac-Moody algebra — an infinite-dimensional Lie algebra with remarkable representation theory connected to modular forms and conformal field theory.

The Gabber-Kac theorem (1981). For a symmetrisable generalised Cartan matrix, the Serre presentation is equivalent to the defining relations of the Kac-Moody algebra (the radical of the standard bilinear form on the free Lie algebra).

Quantum groups. The Serre relations have a $q$-deformed version: $e_i^2{e}_j-(q+q^{-1})e_i{e}_j{e}_i+e_j{e}_i^2=0$ (the quantum Serre relation). This defines the quantum group $U_q(\mathfrak{g})$, a deformation of $U(\mathfrak{g})$ that is central to the theory of quantum integrable systems and knot invariants.

Synthesis. Serre's theorem is the presentation theorem for semisimple Lie algebras; the central insight is that the Cartan matrix provides a complete set of generators and relations, converting the continuous Lie algebra into a finitely presented algebraic object. This pattern appears again in the Coxeter presentation of reflection groups and the Lusztig canonical basis of quantum groups. The bridge is that presentations by generators and relations are the algebraic counterpart of the geometric root system — Serre's theorem identifies the two.

Full proof set [Master]

Proposition ($\mathfrak{sl}(2)$-string length). In any finite-dimensional $\mathfrak{g}(A)$-module, the $\mathfrak{sl}(2{)}_i$-string through a weight vector of weight $\lambda$ has length $\lambda (h_i)$.

Proof. The operators $e_i$ and $f_i$ generate an $\mathfrak{sl}(2{)}_i$-action on the weight space chain. The highest weight of this string is $\lambda (h_i)$ and the lowest is $-\lambda (h_i)$, by the $\mathfrak{sl}(2)$ representation theory. The string length (number of steps from bottom to top) is $\lambda (h_i)+1$. $\square$

Connections [Master]

The Cartan matrix 07.06.19 provides the input to Serre's theorem; each entry $A_{ij}$ determines the exponent in the Serre relations.

Root-space decomposition 07.06.18 gives the root system that the Serre presentation reconstructs; the Serre relations ensure the generated root system matches the original.

The Casimir element 07.06.21 and Weyl complete reducibility 07.06.22 are needed in the proof that $\mathfrak{g}(A)$ is semisimple; the positive-definiteness of the Killing form follows from the properties of the Cartan matrix.

Dynkin diagrams are the graphical encoding of the Cartan matrix, and the Serre presentation can be read directly from the diagram: each node gives a pair of generators, each edge gives a Serre relation.

Bibliography [Master]

@book{serre1966,
  author = {Serre, Jean-Pierre},
  title = {Alg{\`e}bres de Lie semi-simples complexes},
  publisher = {W. A. Benjamin},
  year = {1966}
}

@book{humphreys-lie,
  author = {Humphreys, James E.},
  title = {Introduction to Lie Algebras and Representation Theory},
  publisher = {Springer},
  year = {1972}
}

@book{kac-infinite,
  author = {Kac, Victor G.},
  title = {Infinite-Dimensional Lie Algebras},
  edition = {3},
  publisher = {Cambridge University Press},
  year = {1990}
}

Historical & philosophical context [Master]

Jean-Pierre Serre published his presentation theorem in his 1966 lecture notes ^{[Serre 1966]}, giving a clean algebraic construction of semisimple Lie algebras from their Cartan matrices. Prior to Serre, the construction of Lie algebras from root systems required ad hoc arguments for each classical type.

The Serre presentation opened the door to Kac-Moody theory: Victor Kac and Robert Moody independently discovered in 1968 that the same presentation, applied to indefinite Cartan matrices, produces infinite-dimensional Lie algebras with rich structure. The affine Kac-Moody algebras became central to mathematical physics through their connections to vertex operator algebras, conformal field theory, and the monstrous moonshine conjectures.

The philosophical point is that a semisimple Lie algebra is completely determined by its combinatorial data (the Cartan matrix). The continuous geometry of the Lie group is encoded in the discrete algebra of the Serre relations. This is the deepest expression of the principle that Lie theory translates between the continuous and the discrete.